Parameter estimation of the LDF coefficients for the process using Langmuir-Freundlich

The following parameter estimation is being performed for the breakthrough process employing the Langmuir-Freundlich multilayer isotherm, with the aim of estimating the LDF coefficients for each component in each layer of adsorbent, considering the influence of this parameter in the breakthrough curve.

An initial guess is also required for all the parameters being estimated, as well as, upper and lower bounds for each parameter. These values are given in the following Table 4.4. The values used as initial guess for the LDF coefficients are similar to the ones provided in the literature, once that the closer the initial guesses are to the values, the faster is the parameter estimation procedure.

Table 4.4 – Initial values for the LDF coefficients LDF coefficients, 𝜔

Components Activated carbon Zeolite Lower bound Upper bound

Hydrogen 0.65 0.65 0 1

Methane 0.15 0.13 0 1

Carbon Dioxide 0.03 0.011 0 1

Carbon Monoxide 0.12 0.05 0 1

Nitrogen 0.2 0.1 0 1

The data being used as measured data for the parameter estimation consists in the breakthrough data obtained from the validations made with the Langmuir-Freundlich for the five component mixture and, with a column with two layers, one of activated carbon and the other of zeolite, both with equal lengths (section 4.2.2.2 of this work).

With the aim of turning this process in a more realistic situation, the mentioned data is going to be subject to some treatment through the addition of noise. For this matter, Microsoft

Excel was used and the noise was obtained resorting to the NormInv function. The noise

obtained with the excel function was then added to the molar fraction values from the breakthrough simulations obtained in section 4.2 of this work. Different sets of data were used in order to verify which one produced the finest estimation. With the intention of obtaining the best results possible, the data used as measured data included the breakthrough of all the components existing in the mixture, in order to provide gPROMS® with data describing the behaviour of all the components.

Although a considerably high number of points is being provided to gPROMS® it was considered that with such amount of data a better estimation could be obtained, despite the fact that the simulation time would increase significantly.

4.4.1.1 Parameter Estimation with a variance of 0.05

The first set of data was obtained considering a random probability, a mean equal to zero and standard deviation of 0.05. When introducing the experimental data for the parameter estimation, gPROMS® requires a value for the variance. For the following simulation a constant variance of 0.05 was considered.

Considering that carbon dioxide is the last component to breakthrough (breakthrough time ≅ 4500) the measured data for the present parameter estimation is considered till 𝑡 = 5500 𝑠 with the purpose of having the breakthrough of all components present in the mixture. Figure 4.17 below shows the experimental data introduced in gPROMS® and the major iterations performed by this tool for the hydrogen, in order to obtain the estimated parameters present in Appendix 5 (Table 5.12). Although it is possible to obtain a graphical representation of all iterations performed by gPROMS®, it was agreed to obtain only the main results considering that, with this option, the amount of time required to perform each parameter estimation is reduced when compared with the estimation which provides all the iterations. For this situation, two major iterations were required by gPROMS® to obtain the desired parameters.

Figure 4.17 – Experimental data and major iterations performed by gPROMS® for the parameter estimation

The analysis of these results shows that the final values obtained through the parameter estimation tool are close to the original values present in the respective literature [31]. However, when the analysis is made concerning the confidence intervals, it can be seen that the range of the confidence intervals associated with some of the estimated parameters is

0,5 0,6 0,7 0,8 0,9 1 1,1 0 1000 2000 3000 4000 5000 mol ar fr ac tion ( H2 ) time (s) Experimental data initial iteration 1: Major iteration 2: Major iteration

considerably large. For example, for hydrogen, in both adsorbents, the estimated values are obtained with the highest level of uncertainty when compared with the other components. Relatively to the parameters estimated for the activated carbon layer, the final value obtained for carbon dioxide is the one associated with a smaller uncertainty. All the parameters estimated for the remaining components in the activated carbon layer have a high level of uncertainty. However, the estimated parameters for the zeolite layer exhibit a higher level of certainty when compared with the values from the activated carbon layer.

With the aim of obtaining parameters with a higher level of certainty the final values obtained in the previous simulation were used as an initial guess for a new simulation with the same constant variance. Only a major iteration was required to obtain the new estimations. The results obtained were equal to the ones in the first estimation. The fact that the values obtained were the same as the initial values provided to gPROMS® means that these LDF coefficients are the optimal values that could be obtained by gPROMS® considering the initial guesses provided.

A breakthrough simulation was performed employing the LDF coefficients obtained through the parameter estimation with the aim of comparing the curves with the experimental ones. The analysis of Figure 4.18 shows that the breakthrough curves obtained in this simulation are in accordance with the ones obtained with the LDF coefficients suggested in the literature and with the experimental curves [31].

Figure 4.18 – Comparison of the experimental data with the simulation results employing the LDF coefficients obtained through the parameter estimation

In the previous parameter estimations it was assumed that the lower and upper bond were the same for all the parameters .However, given that the experimental data seen through the literature review present in chapter 2 for mixtures with the same components as the ones being used, an idea of the magnitude of each parameter for each component can be made. Taking this into consideration, a new estimation is going to be performed with the same

0 0,2 0,4 0,6 0,8 1 10 100 1000 mol ar fr ac tion time (s) H2 exp. H2 CH4 exp CH4 CO2 exp. CO2 N2 + CO exp. CO+N2

conditions and initial guesses, however, with different lower and upper bounds for each component. The new lower and upper bounds employed for the following simulation are gathered in Appendix 5 (section 5.1.1).

The parameters for these conditions were obtained after two major iterations and the results and respective confidence intervals are presented in Appendix 5 (Table 5.14). As can be seen in the analysis of these results, gPROMS® didn’t provide confidence intervals for methane in the activated carbon layer. This situation occurs when the parameter being estimated assumes the value of the upper or lower bound, preventing gPROMS® from being able to perform the iterations needed and, therefore, preventing it from providing the results. The comparison of the parameters estimated when considering a smaller interval where these values can change with the results obtained in the estimation with a higher interval of variation shows that the final values obtained differ more from the ones suggested in the literature [31]. However, better confidence intervals were obtained for hydrogen, carbon monoxide and nitrogen in both adsorbents, although the parameters for hydrogen are still estimated with a high level of uncertainty when compared with the remaining components. A problem associated with small intervals, between the lower and upper bounds, relies in the fact that several simulations are stopped due to the value being estimated “hitting” one of the bounds, like happened for methane in the present parameter estimation. For this reason, and also due to the fact that the values obtained with this method provided parameters considerably different from the ones in the literature, in the following simulations concerning the estimation of the LDF coefficients the lower and upper bounds equal to zero and one, respectively, are going to be assumed for all the components in both adsorbents.

According to the component, the magnitude of the molar fraction used as measured data varies. For example, the molar fractions of hydrogen are always high considering that it is the component that is intended to be purified and the molar fractions of carbon dioxide are small due to its molar fraction in the feed mixture. For this reason, an attempt of using a relative constant variance model in the parameter estimation was made, taking into consideration that with this variance model errors are directly proportional to the magnitude of the measured value.

The results for this simulation show that, while the initial iteration resembles with the measured data used as input for the parameter estimation procedure, the major iterations produced by this tool are quite different (see Appendix 5, Figure 5.13) which could explain the lack of sense in the final values obtained from this simulation (Appendix 5, Table 5.14). The analysis of the measured data provided for this estimation shows that the number of observations with lower molar fraction is higher than the number of observations with a high molar fraction. For this reason, the greater effort of gPROMS® is made for the lower molar

fraction values, which explains the difference between the curves associated with the major iterations and the measured data for the higher molar fraction values.

4.4.1.2 Parameter Estimation with a variance of 0.01

With the aim of verifying if better estimated parameters were obtained, it was agreed to decrease the standard deviation associated with the noise added to the breakthrough data. For the following simulations the breakthrough data was obtained, again, with the NormInv function from Microsoft Excel, considering now a mean of zero and a standard deviation of 0.01. The first simulation is performed assuming a lower bound of zero and an upper bound equal to 1 for all components. A constant variance model of 0.01 was considered for the following parameter estimation.

A total of 6 major iterations were required to estimate the LDF parameters for the considered situation, as can be observe in Figure 4.19, as well as the experimental data used for the considered simulation.

The first effect of decreasing the variance and the standard deviation of the noise of the “experimental” values is the increase in the number of major iterations required to obtain the desired parameters. While for a constant variance of 0.05 two major iterations were required, for a constant variance of 0.01 a total of six major iterations was required.

Figure 4.19 - Experimental data and major iterations performed by gPROMS® for the parameter estimation for a constant variance of 0.01

The comparison of the results present in Appendix 5 (section 5.1.2, Table 5.15) with the results showed in Appendix 5 (section 5.1.1, Table 5.12) shows that the reduction in the value attributed to the variance provides estimated parameters with a higher level of certainty. For example, the analysis of the confidence intervals associated with the estimation of the LDF coefficient for hydrogen shows a higher level of certainty of the estimation when the variance

0,5 0,6 0,7 0,8 0,9 1 0 2000 4000 6000 mol ar fr ac tion ( H2 ) time (s) Experimental data initial iteration 1: Major iteration 2: Major iteration 3: Major iteration 4: Major iteration 5: Major iteration 6: Major iteration

is 0.01. However, the estimation of the hydrogen LDF coefficients is the one with the higher level of uncertainty. When comparing the estimated values per layer of adsorbent, the data obtained from the estimation shows that predictions with a higher level of certainty are obtained for the zeolite layer.

The final values obtained are now going to be used as initial guess for a new simulation with the aim to investigate if the results are again the same as it happened with the simulation for a constant variance of 0.05 or if better estimations are being obtained.

The use of the final values from the simulation present in Figure 4.19 as initial guesses for a new parameter estimation showed that the values obtained were not the optimal values, as happened for the constant variance of 0.05. gPROMS® performed 7 major iterations and the process was interrupted due to a numeric error. With the aim of avoiding this error the bounds of molar and mass fraction as well as the bounds of velocity were changed. The valves coefficients were also increased considering that gPROMS® provided information related with problems in maintaining the desired pressure inside the adsorption bed.

Six major iterations were again required to obtained new values which lead us to conclude that the final results from the first simulation with a constant variance of 0.01 were not the optimal ones.

The results obtained for the previously mentioned parameter estimation shows that gPROMS® provides estimations with high levels of uncertainty. The LDF coefficients of hydrogen are the ones with the higher level of uncertainty, which could be related with the fact that this is the less strongly adsorbed, and therefore, the effect of the LDF coefficient of this component in the control variable is weaker when compared with the other components. Due to this reason, it is harder to gPROMS® to provide a better estimation.

From all the estimations above the parameter estimated with a higher level of certainty was the LDF coefficient for carbon dioxide in the zeolite when considering a constant variance of 0.01. This component is the one which adsorbs more strongly in the considered adsorbent which could explain the fact of this being the better estimation obtained, due to the fact that a slightly change in the parameter being estimated affects strongly the control variable.